NEW YORK—Mayor Michael Bloomberg has started the new year with his reputation as a first-class manager suddenly in question after a post-Christmas snowstorm left New York paralyzed for days.

Critics of Mr. Bloomberg say his administration's sluggish response to the Dec. 26-27 blizzard—coming on the heels of a scandal involving the theft of $80 million from the city's long-delayed computerized payroll system—may have damaged the mayor's legacy.

A NY1-Marist poll conducted Wednesday showed Mr. Bloomberg with a 37% approval rating, his lowest since taking office in January 2002. More than seven in 10 disapproved of the mayor's handling of last week's snow removal.

"Obviously, he's concerned about his legacy, as anyone would be, and this [snowstorm] will have a big effect on that," said City Council member Peter Vallone, a Democrat who crossed party lines to endorse the Republican-turned-independent Mr. Bloomberg in 2009.

The mayor said last week the city's response to the storm was "inadequate" and "unacceptable." But he added that there were an "awful lot of good numbers out there" about the city's progress, and he was proud overall of his administration's accomplishments.

In his nine years as chief executive of the nation's largest city, the mayor has earned plaudits for solving multibillion dollar budget deficits, taking control of the city's school system and spearheading a number of ambitious public-health initiatives, including a popular citywide ban on smoking in restaurants and bars.

Mr. Bloomberg, a 68-year-old billionaire, flirted with the possibility of running for president in 2008, but decided he didn't have a chance as a third-party candidate. He has said repeatedly he isn't considering a presidential bid in 2012. Some supporters, though, have launched a campaign to draft him.

But a series of missteps in recent months—most notably the city's poor response to the snowstorm—have left even his strongest supporters heaping criticism his way, leaving the Bloomberg brand bruised.

During the heat of the snow crisis last week, Mr. Vallone demanded the mayor stop asserting the city was winning the war on snow when clearly the administration wasn't. The mayor ultimately changed his tone from defiant and defensive to conciliatory and apologetic, and the administration was on high alert for a much smaller snowstorm that began Friday.

In the NY1-Marist poll, when asked about Mr. Bloomberg's legacy, 12% of respondents said they thought he would be remembered as one of the best mayors; 27% above average; 35% about average; 15% below average; and 10% one of the worst.

In recent months, the mayor has been embarrassed by the arrest of six people on charges of stealing from the payroll system, known as CityTime. The project, once described by the mayor as a "disaster," was already over-budget, and critics said he failed to properly police it. The mayor also sparked criticism with his selection of the new schools chancellor, Cathie Black, a former publishing executive with no experience in city schools.

With homicides in New York slightly increasing in the past year and some advocates accusing the mayor of failing to do enough to help the working poor, the first year of the mayor's third term had already garnered mixed reviews. Some observers note the mayor—who lost several key advisers last year—could be suffering from inexperience in his inner circle.

Former New York City Mayor Ed Koch, a strong supporter of Mr. Bloomberg, acknowledged the recent missteps but defended the mayor.

"Yes, you are going to be criticized. But in the examination of his nine years as mayor, he's done a terrific job," Mr. Koch said. "If he wanted to be president, we'd be lucky if he was."

Aides to the mayor pointed out that the poor response to the snowstorm was reminiscent of the city's poor response to a power blackout in Queens in 2006. The mayor took responsibility for the slow response there, learned from that incident and was re-elected, they said.

"He didn't run for office in the first place just so he could leave with an enormous amount of political capital," Bloomberg spokesman Stu Loeser said. "He asked to serve for a third term precisely because the city had tough problems, and he felt he could help New York create jobs and tackle its fiscal problems."

Some critics, though, have begun to compare Mr. Bloomberg with former California Gov. Arnold Schwarzenegger, who left office this week with mixed reviews. Three years ago, Time Magazine lauded Messrs. Bloomberg and Schwarzenegger as "The New Action Heroes." But time appears to have hurt both men's images.

Manhattan Borough President Scott Stringer, a potential mayoral candidate when Mr. Bloomberg's term ends in three years, said he believed the mayor's administration was "at a crossroads."

"If he wants to be the greatest mayor ever, he has to look at making some real changes," Mr. Stringer said. "He doesn't want to go out the door deflated and defeated à la Schwarzenegger. The final judgment will be three years from now."